Reynolds number capillaries

La.mina.r Flow Elements. Each of the previously discussed differential-pressure meters exhibits a square root relationship between differential pressure and flow there is one type that does not. Laminar flow meters use a series of capillary tubes, roUed metal, or sintered elements to divide the flow conduit into innumerable small passages. These passages are made small enough that the Reynolds number in each is kept below 2000 for all operating conditions. Under these conditions, the pressure drop is a measure of the viscous drag and is linear with flow rate as shown by the PoiseuiHe equation for capilary flow ... 

In creeping flow with the inertia term neglected, the velocity distribution rapidly reaches a steady value after a distance of r0 inside a capillary tube. At this stage the velocity distribution showed the typical parabolic shape characteristic of a Poiseuille flow. In the case of inviscid flow where inertia is the predominant term, it takes typically (depending on the Reynolds number) a distance of 20 to 50 diameters for the flow to be fully developed (Fig. 34). With the short capillary section ( 4r0) in the present design, the velocity front remains essentially unperturbed and the velocity along the symmetry axis, i.e. vx (y = 0), is identical to v0. 

Chain degradation in turbulent flow has been frequently reported in conjunction with drag reduction and in simple shear flow at high Reynolds numbers [187], Using poly(decyl methacrylate) under conditions of turbulent flow in a capillary tube, Muller and Klein observed that the hydrodynamic volume, [r ] M, is the determining factor for the degradation rate in various solvents and at various polymer concentrations [188], The initial MWD of the polymers used in their experiments are, however, too broad (Mw/Iiln = 5 ) to allow for a precise... 

For axial capillary flow in the z direction the Reynolds number, Re = vzmaxI/v = inertial force/viscous force , characterizes the flow in terms of the kinematic viscosity v the average axial velocity, vzmax, and capillary cross sectional length scale l by indicating the magnitude of the inertial terms on the left-hand side of Eq. (5.1.5). In capillary systems for Re < 2000, flow is laminar, only the axial component of the velocity vector is present and the velocity is rectilinear, i.e., depends only on the cross sectional coordinates not the axial position, v= [0,0, vz(x,y). In turbulent flow with Re > 2000 or flows which exhibit hydrodynamic instabilities, the non-linear inertial term generates complexity in the flow such that in a steady state v= [vx(x,y,z), vy(x,y,z), vz(x,y,z). ... 

It is important that we know at what Reynolds number our instrumental configurations give turbulent flow and work below this figure or we will think that shear thickening is occurring A figure of Re < 3000 to 10,000 is usually satisfactory for cone and plates or capillary viscometers, but values as low as 300 may be the maximum for some cup and bob units. 

The flow of oxygen through the inner capillary of the burner (Fig. 1 b) is laminar. The estimated Reynolds number in this region for 1000 bar (Fig. 4) is about 200, much below the critical number for turbulence. This is also true for the other pressures investigated. The flames can clearly be considered as diffusion flames. Because of their conical shape the conventional simplified treatment of laminar diffusion flames can be applied [16 — 18]. According to Burke and... 

The flow rate of blood through the heart is approximately 4-5lmin for adults. The typical mean blood velocity through the aorta (which is the largest artery with a diameter of 2-3 cm), when pumped from the left heart, is approximately 25 cm s"i (mean) the maximum velocity is approximately 60 cm s h The Reynolds number for the maximum velocity is about 3000. In general, the blood flow through arteries and veins is laminar in nature. In capillaries, the typical blood velocity is 0.5-1 mm s , and the Reynolds number is on the order of 0.001. 

Washing. As ice moves upward through the column, it behaves as an unconsolidated porous medium and carries brine with it, held by viscous and capillary forces. The flow is entirely laminar, because the Reynolds number based on particle diameter is always less than 0.05. The brine is carried thus at the surface and in the fillets between the particles. The downward flowing wash water moves between the particles and mixes with the brine mainly by molecular diffusion. Salt will diffuse from the brine to the wash water. 

The conversion of the throughput qpV into volume flow yields the values listed in Table 1. Since in the capillaries laminar flow prevails (Reynolds number RE 2,300), the pressure drop along the capillaries follows Hagen-Poiseuille s equation ... 

Monophasic fluid flow in capillary-scale ducts is characterized by a low Reynolds number, the flow in capillary-scale microreactors is generally laminar and transport... 

The Reynolds numbers for the flow of a molten metal along a capillary braze gap are usually less than 1000 as will be shown later, and the theoretical laminar flow rates for such configurations have been calculated by Milner (1958) for both horizontal and vertical joints. He regarded the flow into a horizontal joint induced by capillary attraction as being impeded only by viscous drag, and derived a simple parabolic expression to describe such behaviour,... 

The rheology of suspensions generally differs from fluids as a result of the hydrodynamic forces acting on the particles. The following figure illustrates this behavior in a print head. Since flows in print heads are in the range of low Reynolds numbers (Re 1-10 during drop formation in an ink channel with 350 m diameter), the velocity profile within (circular) capillaries is parabolic. This is indicated in Fig. 1. 

For each gas (including air), calculate the maximum value of the Reynolds number from Eq. (11) and verily that it is below 1000. Use Eq. (12) to verily that U, the length of the transition region, is small compared with the length L of the capillary. Estimate the largest value of V from Eq. (5) and compare it with the molecular rms speed u and mean speed c. 

Archimedes number Bingham number Bingham Reynolds number Blake number Bond number Capillary number Cauchy number Cavitation number Dean number Deborah number Drag coefficient Elasticity number Euler number Fanning friction factor Froude number Densometric Froude number Hedstrom number Hodgson number Mach number Newton number Ohnesorge number Peclet number Pipeline parameter... 

In coaxial mixers, the two reactant flnids are initially in an onter and an inner capillary. The two flnids run parallel over a short distance toward a Pt sphere. At this stage they do not mix because of the low Reynolds number of the flow, which is essentially laminar (Figure 2b). The two fluids meet at a small (50 250 im) platinum sphere, which is positioned at the tapered end of the onter capillary. The two flnids are forced between the small space (5-10 tm) between the Pt sphere and the outer capillary where they are accelerated creating a small zone of high turbulence Re 30 000). Withont the sphere, the channel dimensions and flow rates would yield a calculated Re of 1500, which is insufficient for mixing. 

The viscosity as calculated according to Eq. (44) is meaningful only if the flow is laminar. For a capillary flow the rate of flow should not exceed a critical velocity which can be determined from its Reynolds number (Rk) ... 

If gravitational settling can be neglected and if the droplet Reynolds number Re = payout 9s is small, then the droplet deformation and possible breakup in the flow are controlled by two dimensionless groups, namely the ratio of viscous to capillary forces, or capillary number... 

Aul and Olbricht [4] reported the results of an experimental study of low-Reynolds number, pressure-driven core-annular flow in a straight capillary tube. The annular film was thin compared to the radius of the tube, and the viscosity of the film fluid was much larger than the viscosity of the core fluid. The photographs showed that the film was... 

Aul, R.W. and Olbricht, W.L., Stability of a Thin Annular FUm in Pressure Driven, Low-Reynolds-Number Flow Through a Capillary, Journal of Fluid Mechanics, 1990, 215, 585-599. 

In the foregoing discussion, the assumption is that each phase is transported in its own percolating network by a pressure-driven flow mechanism. This is the generally accepted view of multiphase flow in subsurface applications, and is certainly true at low values of the capillary number Ca = vfi/a). However, blob mobilization is a dominant form of transport in many unit operations in chemical engineering, where the capillary number and Reynolds number are higher. In these cases, specialized correlations for multiphase flow should be used. 

Similarly, the Capillary number, Ca, determining the ratio between the viscous-and surface tension forces, is related to the Weber number, We, and particle Reynolds number, Rep. That is, the Capillary number can be expressed as. 

Yin et al. (2006) qnalitatively showed this mechanism by solving relevant flow equations nnmericaUy. Xia et al. (2008) also developed a simplified pore scale model to describe polymer flow. The numerical solutions from Xia et al. have verified the proposed mechanism. Figure 6.22 shows the velocity contours of a Newtonian fluid with Weissenberg number (We) = 0 and a viscoelastic fluid with We = 0.35 in a flow channel with a dead end when the Reynolds number (Re) = 0.001. We can see that the velocity (m/day) of the viscoelastic fluid is higher than that of the Newtonian fluid at the same position of the dead end. This pulling mechanism also works in the case shown in Figure 6.20c, where the residual oil is trapped at the pore throats by capillary force. 

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